Prof. Rohan C Abeyaratne

Certain continuum mechanical models of solids allow for the occurrence of surfaces of strain discontinuity when the body is finitely deformed; across such a surface, the stress and strain fields suffer finite jump discontinuities even though the traction and displacement fields remain continuous. [In certain respects, this is analogous to the well-known phenomenon associated with high speed flows in Gas Dynamics, where one often finds smooth flow fields separated by shock waves].

One area to which such a model finds application is the continuum mechanical description of a solid in equilibrium with more than one "phase" present; in this setting, a surface of strain discontinuity corresponds to the boundary between two distinct phases of the material. As the body deforms, the phase boundary propagates through the body, and material in one phase transforms to another as it crosses this moving surface. The phenomenon of phase transformation is the underlying reason, for example, for the unusual behavior of shape-memory metals and also for the toughening effect observed in certain ceramic composites.

If equilibrium states of the body correspond to global minima of an appropriate energy functional, then the "driving force" on the phase boundary vanishes, and boundary-value problems describing equilibrium configurations are well-posed. On the other hand, if an equilibrium state is merely a local minimizer of energy, corresponding to a metastable state, the driving force does not vanish; quasi-static motions involving such metastable equilibrium states are therefore dissipative and are not well-posed. Consequently, in order for the mathematical models describing the response of such materials to be complete, they must be broadened so as to include additional constitutive information pertaining to the kinetics which govern the evolution of phase boundaries.

In recent studies we have examined the nature of the driving force on a phase boundary in circumstances that involve both thermal and inertial effects and in a setting which is not restricted to any particular constitutive law. We have also studied the kinetics governing the motion of a phase boundary when its velocity is given by a material-dependent function of the driving force. We have determined the particular kinetic function which arises when the underlying micro-mechanical processes are controlled by viscosity and non-locality. A model based on thermal activation theory has also been developed.

The presence of inertial effects lead to some new phenomena. In particular, motions of a body can now involve shock waves, which themselves are surfaces across which the stress and strain fields are discontinuous, and so, one now has to carefully distinguish between this type of discontinuity and a phase boundary. It is well-known that dynamical processes under these circumstances lead to non-unique solutions and that various "admissibility conditions" have been proposed in the mathematical literature in order to secure uniqueness. We have recently explored the relationship between such mathematical admissibility criteria and the more physically based approach based on kinetic relations. While the kinetics govern the evolution of an existing phase boundary, a separate nucleation criterion is required to signal the initiation of the transformation. Appropriate ways for including this effect in the continuum model are not well understood.

Recently we applied these ideas to study the twinning of a Cu-Al-Ni shape memory alloy. In the course of this investigation we have begun to develop some new results concerning the homogenized response of a material with evolving microstructure.

Current research is concerned with further questions related to the issues outlined above. The approach is primarily analytical, and is focussed on fundamental concepts and issues. The predictions of the model during various thermo-mechanical loading processes are being studied. Some numerical work, as well as experiments are to be carried out. In particular, the following are some of the questions to be explored: Effect of temperature, Kinetic relations and their micro-mechanical basis, Three-dimensional effects with attention on deformations involving fine mixtures, and Effect of surface structure.